Question

the mass of the glider (m) and the effective
spring constant (k). With a balance you
measure the mass of the glider to be m = 0.287 kg. [We ignore
the mass of the springs in this lab.] To determine k, consider the setup
described in Activity 1-2. The procedure is to measure the position
x of the
glider as force F is applied to the
glider (the force probe both applies the force that stretches the
spring and measures it). In the actual investigation you will be
using Data Studio to create a graph, then fit a straight line to
the graph. Here we provide you with a table:

d
(m)

F
(N)

0.3899

0.212

0.4318

0.434

0.4738

0.657

0.5157

0.879

a) Use linear least squares to find the slope of this force
vs. position data. From this slope, calculate the effective spring
constant k:
N/m

b) Calculate the natural frequency f0
you should expect for this mechanical oscillator:
Hz

Investigation 2 concerns the free oscillations of a damped
harmonic oscillator. The setup is exactly the same as for
Investigation 1 with the exception that some magnets are placed on
the skirts of the glider to provide an eddy current damping force
(and there is no force probe pulling on the cart!).

IMPORTANT NOTE: These magnets were actually attached to
the glider in Investigation 1, but they were positioned so as to
NOT generate any damping. This means that the mass of the glider is
unchanged during the entire lab.

You pull the glider to one side from its equilibrium position,
release it from rest, and record the position vs. time of the
decaying oscillation. Your data will be somewhat like that shown in
the first figure of Investigation 2 in the manual. Using the
SmartTool, the following peak times and amplitudes were
measured:

Peak

t
(s)

A
(m)

0

0.0790

0.2527

1

1.5422

0.1988

2

3.0054

0.1564

3

4.4686

0.1230

4

5.9317

0.0968

c) Use the time difference between the first and last peaks to
calculate the frequency of the oscillation:
Hz

We could try to use the calculated frequency of the undamped
system and the measured frequency of the damped system to obtain
the decay time constant ?, but this frequency
shift is very small and it is difficult to measure the �peak times�
with great precision. A better way to obtain ? is to look at the decay
of the peak amplitudes over time. Read the first couple of pages of
Investigation 2 in the lab manual if you are not sure what the
variable ?
is.

d) Again using the first and last peaks, calculate ?:
s

e) Use this calculated ? along with the mass of
the glider m
to obtain the damping coefficient b (the coefficient of the
velocity term in the equation of motion, Equation 13):
N�s/m

In Investigation 3, you measure the amplitude A of oscillation as a
function of frequency for the driven oscillator. The spring support
at one end of the air track is driven at frequency f and the amplitude
A(f)
of the motion of the glider is recorded as a function of frequency.
Figure 11.5 in the manual shows the ratio A/A0
for various values of Q (defined by Equation
11.30), where A0
is the amplitude measured at a frequency very small compared to the
natural frequency f0.

f) Use the undamped frequency (?0),
the mass of the glider (m), and the damping
coefficient (b) to predict
Q:

g) If the amplitude of the system driven at a frequency MUCH lower
than the resonant frequency is 0.0256 m, predict the amplitude of
the system when the driving frequency is raised to the resonant
frequency:
m